Mark Harper, the immigration minister, has said it would not be “sensible” to try and predict how many Romanians and Bulgarians are expected come into the country after getting the right to live and work in the UK.

He insisted that the Government is not doing any research on their likely impact because “there are so many variables”.

Chris Bryant, the Labour MP for Rhondda, accused Mr Harper of trying to “cover up” details of an official analysis understood to have been seen by the Government.

Eric Pickles, the Communities Secretary, has admitted that he has seen figures on the potential number of immigrants, but said that he was not willing to divulge them.

The Daily Telegraph last week disclosed that ministers are refusing to release the analysis because Whitehall officials believe there will only be a minimal influx.

A discussion paper presented at a recent Cabinet committee suggested that the number emigrating to Britain when controls are lifted was only likely to run into the thousands.

Ministers are concerned about releasing the research, which they believe will be compared with a prediction that only 13,000 would move to Britain from Poland and other eastern European countries after 2004.

More than one million eastern Europeans then arrived in one of the biggest waves of immigration seen in the UK.

Twenty nine million Bulgarians and Romanians will gain the right to live and work unrestricted in Britain in 2014 under European “freedom of movement” rules.

In an interview with BBC Radio 5 Live Mr Harper said the Government is “not going to start speculating about numbers”.

He said the Government migration advisory committee “made it very clear” that making any predictions “wasn’t a very sensible thing to do”.

“There are so many variables,” Mr Harper said. “There are eight other European countries which have transitional controls in place [and] will remove them at the same time.

“What the Government is focussing on is…the things that might encourage people to come to the United Kingdom rather than other European countries.”

Asked whether the Government is doing any research on probable numbers preparing to come to the UK Mr Harper said: “No we’re not.”

“Trying to do that is not a very sensible thing to do,” he said. “The previous government made a forecast which turned out to be completely wrong.

“The problem with this is very simple. You make a forecast and it ends up being based on a whole set of assumptions. You don’t know exactly what’s going to happen.”

Chris Bryant, the shadow immigration minister, said that he believes the figures now “have to be released”.

“It certainly does feel as if they are trying to cover something up,” Mr Bryant said. “They are hiding something from us. Eric Pickles clearly said there were figures.”

“When it comes to immigration, the government should be as open and honest as possible.

“By refusing to release these figures and now denying their existence the Government will lose the trust of the British people.”

A report from MigrationWatch, a think tank, has claimed that 50,000 people a year will arrive in the UK until 2019 when the transitional controls are lifted.